Matrix, mathematical term.In mathematics, a matrix is a rectangular arraycomplexorreal numberaggregate[1], originally fromEquationsOfcoefficientandconstantConstitutedmatrix。This concept was developed by British mathematicians in the 19th centuryKellyFirst, put forward.
The main branch of numerical analysis is devoted to the development of effective algorithms for matrix computation, which has been a subject for several centuries and is an expanding research field.The matrix decomposition method simplifies the theoretical and practical calculation.The algorithm customized for specific matrix structure (such as sparse matrix and near angle matrix) speeds up the calculation in finite element method and other calculations.The infinite matrix occurs in planetary theory and atomic theory.A simple example of an infinite matrix represents a functiontaylor series Matrix of derivative operator of[3]
The study of matrix has a long history,Latin squareHe Magic Square has been studied in prehistoric times.
Arthur Kelly, Founder of Matrix Theory[4]
In mathematics, a matrix is a rectangular arraycomplexorreal numberaggregate[1], originally fromEquationsOfcoefficientandconstantConstitutedmatrix。This concept was developed by British mathematicians in the 19th centuryKellyFirst, put forward.As a solutionlinear equationMatrix has a long history.The book was first written in the early Eastern Han Dynasty《Chapter Nine Arithmetic》Medium, separated withcoefficientLegal representationLinear equations, obtained itsAugmented matrix。stayEliminationTo multiply a line by a nonzeroreal number, subtracting one line from another, which is equivalent to theElementary transformation。But at that time, there was no matrix concept understood today. Although it was the same form as the existing matrix, it was only used as the standard representation and treatment of linear equations at that time.
The matrix formally appears as the research object in mathematicsdeterminantAfter the development of research.Logically, the concept of matrix is prior to determinant, but in actual history it is just the opposite.Japanese mathematicianGuan Xiaohe(1683) andCalculusOne of the discoverers ofGottfried William Leibniz(1693) established independently almost simultaneouslyDeterminant theory。Followed by determinant as solutionLinear equationsThe tools of are gradually developed.In 1750,Gabriel KramerFoundCramer's rule [5]。
James Joseph Sylvester
The concept of matrix came into being in the 19th century.In the 1800s,GaussianandWilliam JordanEstablishedGauss Jordan elimination method。In 1844,GermanymathematicianFerdinand Eisenstein(F. Eisenstein) discussed "transformation" (matrix) and its product.In 1850, British mathematicianJames Joseph Sylvester(James Joseph Sylvester) First uses the word matrix.
britainmathematicianArthur Kelly is generally recognized as a matrix theoristfounder。He began to treat the matrix as independentmathematicsWhen studying objects, many properties related to matrices have been found in the study of determinants, which makes Kelly think that the introduction of matrices is very natural.He said, "I definitely didn't passQuaternionAnd obtain the concept of matrix;It is either directly derived from the concept of determinant, or as an expressionLinear equations"Since 1858, he has published a series of special papers on matrices, such as Research Report on Matrix Theory, and studied the operation law, inverse andTranspositionandCharacteristic polynomialEquation.KellyThe Kelley Hamilton theorem is also proposed, and the case of 3 × 3 matrix is verified. It is also said that further proof is unnecessary.HamiltonThe case of 4 × 4 matrix is proved, and the proof in general is made by German mathematiciansFrobenius(F.G. Frohenius) in 1898[5]。
1854FrancemathematicianHermit(C. Hermite) used“Orthogonal matrix”This term, but his official definition was not published by Philobenius until 1878.In 1879, Ferobenius introduced the matrixRankThe concept of.So far, the matrix system has been basically established.
The study of infinite dimensional matrix began in 1884.Poincare After two articles that loosely used the theory of infinite dimensional matrix and determinant, I began to study this aspect.In 1906,HilbertThe introduction of infinite quadratic form (equivalent to infinite dimensional matrix) to study the integral equation greatly promotes the study of infinite dimensional matrix.On this basis,Schmitz Herringer and Triplitz developed the operator theory, and the infinite dimensional matrix became the researchfunctionA powerful tool for space operators[6]。
MatrixconceptIt was first seen in Chinese in 1922.In 1922,Cheng TingxiIn an introductory article, the matrix is translated as "vertical and horizontal matrix".In 1925, the computational terminology review group of the Scientific Terminology Review Association《science》In the list of approved terms published in the fourth issue of Volume 10, the matrix is translated as "matrix",square matrix It is translated as "square matrix", while various matrices such as "orthogonal matrix"“Adjoint matrix”The "matrix" in is translated as "square matrix".In 1935,Chinese Mathematics SocietyAfter review,Ministry of Education of the Republic of ChinaIn the approved "Mathematical Nouns" (and "all colleges and universities across the country are instructed to follow the rules and regulations"), "matrix" appears as a translation name for the first time.In 1938, Cao Huiqun was entrusted by the Scientific Terminology Review Conference toMathematical nounIn the revised Glossary of Arithmetic Terms, the proper translation is "rectangular array".Prepared after the founding of the People's Republic of China《Mathematical noun》In the Chinese version, the translation is "matrix".In 1993, the Chinese Natural Science Terminology Commission announced《Mathematical noun》In Chinese, "Matrix" has been officially translated and has been used up to now.
definition
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By m × n number aijThe number table of m rows and n columns is called the matrix of m rows and n columns, or m × n matrix for short.Recorded as:
The number m × n is called matrixAElement of, referred to as element, number aijIn matrixAIs called matrixA(i, j) element of, with the number aijThe matrix of (i, j) element can be recorded as (aij）Or (aij)m × n, m × n matrixAAlso recorded asAmn。
The element isreal numberThe matrix of is calledReal matrix, element iscomplexThe matrix of is calledComplex matrix。The matrix whose number of rows and columns is equal to n is called n-order matrix or n-order square matrix[7]。
Basic operation
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Matrix operation inScientific computingVery important[8]The basic operations of matrix include matrix addition, subtraction, number multiplication, transpose, conjugate and conjugate transpose[1][9]。
addition
The addition of a matrix satisfies the following operation laws（A，B，CAre homomorphic matrices):
It should be noted that only homomorphic matrices can be added[10]。
subtraction
Number multiplication
The matrix multiplication satisfies the following operation laws:
Addition and subtraction of matrices and number multiplication of matrices; linear operation of matrices[7]。
Transposition
The matrix produced by exchanging the rows and columns of matrix A is called the transposed matrix of A（
）[8]This process is called matrix transposition
The transpose of the matrix satisfies the following operation laws:
conjugate
The conjugate definition of a matrix is:
The conjugation of a 2 × 2 complex matrix (the real part is unchanged, and the imaginary part is negative) is as follows[11]：
be
Conjugate transpose
The conjugate transpose of matrix is defined as:
, can also be written as:
Or as
。The conjugate transposition of a 2 × 2 complex matrix is as follows:
Multiplication of two matrices only when the first matrixANumber of columns and another matrixBCan only be defined when the number of rows of is equal.asAyesm×nMatrix andByesn×pMatrix, their productCIs am×pmatrix
, one of its elements:
And record this product as:
[8].
For example:
The matrix multiplication satisfies the following operation laws:
, thenAbyPositive semidefinite matrix, ifANeither semi positive definite nor semi negative definite, thenAbyIndefinite matrix[15]。The positive definiteness of symmetric matrix is closely related to its eigenvalue.A matrix is positive definite if and only if itfeaturesValues are positive[1]。
Matrix decomposition is to decompose a matrix into the sum or product of several matrices that are relatively simple or have certain characteristics[13]The matrix decomposition methods generally include triangular decomposition, spectral decompositionsingular value decomposition , full rank decomposition, etc.
Trigonometric decomposition
set up
, then A can be uniquely decomposed intoA=UoneR ,amongUoneIs a unitary matrix，RIs the triangular complex matrix on the main line，orACan be uniquely decomposed intoLIs the triangular complex matrix on the main line，Is a unitary matrix
，bringA=FG，Is calledAA full rank decomposition[18]。
LUP decomposition
LUPThe idea of decomposition is to find threen×nmatrixL,U,P, satisfied
。Where L is a unit lower triangular matrix, U is a unit upper triangular matrix, and P is a permutation matrix.But the matrix satisfying the decomposition conditionL,U,PCalled one of matrix ALUPdecompose[19]。
Special category
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symmetric matrix
staylinear algebraIn, the symmetric matrix is aSquare matrix, whose transpose matrix is equal to itself[8]。I.e
。For example:
。
Hermitian matrix
A square complex valued matrix
Is called Hermitian matrix, ifA=AHThat is, its element
In other words, Hermitian matrix is a complex conjugate symmetric matrix[1]。
For a real valued matrix, Hermitian matrix is equivalent to symmetric matrix.
Orthogonal matrix
A Real Square Matrix
It is called orthogonal matrix, if
。
Unitary matrix
A complex valued square matrix
It is called unitary matrix, if
。
Banded matrix
matrix
, if the matrix satisfies condition aij=0, | i-j |>k, then the matrixAIt can be called banded matrix[20]。
Triangular matrix
staylinear algebraIn, the triangular matrix isSquare matrixIt is a kind of the triangle shaped non-zero coefficient.Triangular matrix is divided into upper triangular matrix and lower triangular matrix.if
, then
The matrix of is called upper triangular matrix[8], if
, then
The matrix of is called lower triangular matrix[8]。Triangular matrix can be regarded as a simplified case of general square matrix.
be calledAThe coincidence matrix of.Where linear transformation
Is called consistent transformation[1]。
Vandermonde matrix
Vandermonde matrix（vandermonde matrix ）The name of is derived from the name of Alexandre Th é ophile Vandermonde, and the Vandermonde matrix is a column representingGeometric seriesRelationalmatrix[1]。
For example:
Or in theiLine NojColumn relationship writing:
Hadamard matrix
Hadamard matrix（Adama matrix）Is amatrix, each element is+1 or − 1, and each line is orthogonal to each other[17]。
At this time, all elements on non diagonal lines are 0[8]The matrix at this time is called diagonal matrix.
Block matrix
A block matrix is to divide the matrix into smaller matrices, which are called sub blocks[21]。For example:
The matrix can be divided into four2×2Matrix of:
The partitioned matrix can be written as follows:
Jacobian matrix
Jacobian matrix is the first order of functionpartial derivativeA matrix arranged in a certain way.
It can be expressed as follows:
Rotation matrix
A rotation matrix is a matrix that changes the direction of a vector but does not change its size when multiplied by a vector.The rotation matrix does not include inversion, which can change the right-handed coordinate system to the left-handed coordinate system or vice versa.All rotations plus inversion form a set of orthogonal matrices.
The rotation matrix is studied by the world famous lottery expert and Australian mathematician Dietrov. It can help you lock your favorite number and improve your chances of winning the lottery.First, you need to select some numbers, and then use a certain rotation matrix to fill the numbers you select into the corresponding positions.If some of the numbers you choose are the same as the lottery number, you will definitely win a certain prize.Of course, using this rotation matrix, you can get the maximum benefit with the minimum cost, and it is far less than the cost of compound betting.
The principle of rotation matrix is mathematically related to a combination design: covering design.Covering design, filling design, steiner system, t-design are all combinatorial optimization problems in discrete mathematics.They address how to combine elements in a collection to meet a specific requirement.
The norm of matrix mainly includes three main types: induced norm, element form norm and Schatten norm[13]。
If mapping
Meet the following requirements:
The mapping is called
The matrix norm on the.
Induced norm
The induced norm is also called
On matrix spaceOperator norm(operator norm), defined as:[22]
The commonly used induction norm is p-norm:
The p norm is also known as the Minkowski p norm or
Norms.In particular, when
When, the corresponding induction norms are[23]
Element formal norm
take
The matrix is arranged as a column
And then use the definition of vector norm to get the element form norm of the matrix[24], the formula is as follows:
Schatten norm
The Schatten norm is a matrixSingular valueThe defined norm is defined as:
among
Is the singular value of the corresponding matrix[25]。
application
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image processing
In image processing, the affine transformation of an image can generally be expressed in the form of multiplying an affine matrix and an original image[26], for example,
This represents a linear transformation followed by a translation.
Linear transformation and symmetry
Linear transformation and its correspondingsymmetric, in modern timesphysicsPlays an important role in.For example, inQuantum field theoryMedium,Elementary particleIs based on the special theory of relativityLorentz groupIt means that, specifically, they areSpinor groupPerformance.InclusionPauli matrixAnd more generalDirac matrices The specific expression ofFermionThe physical description of fermions is an indispensable part, and the performance of fermions can be usedSpinorTo express.Describe the three lightestquarkYou need to use an internalSpecial unitary groupSU (3);Physicists will use a simpler matrix expression when calculating, calledGelman matrix, this matrix is also used as SU (3)Gauge groupThe modern description of strong nuclear forceQuantum chromodynamicsThe foundation of is SU (3).alsoKabibo Kobayashi Yichuan Matrix（CKM matrix）: onWeak interactionThe important basic quark state in is different from the quark state with different mass between the specified particles, but the relationship between them is linear, which is expressed by the CKM matrix.
Linear combination of quantum states
In 1925, Heisenberg proposed the firstquantum mechanicsIn the model, the infinite dimensional matrix is used to represent the operator acting on the quantum state in the theory.This is done inMatrix mechanicsIt can also be seen in.for exampledensity matrix It is used to describe "pure" in quantum systemquantum stateThe "mixed" quantum state represented by linear combination of[27]。
The other matrix is an important tool used to describe the scattering experiments that form the cornerstone of experimental particle physics.When particles are inacceleratorThe particles that had not interacted with each other entered the action area of other particles in the high-speed movement, and the momentum changed to form a series of new particles.The collision can be interpreted as the scalar product of the linear combination of the result particle state and the incident particle state.The linear combination can be expressed as a matrix, calledS-matrix, in which all possible interactions between particles are recorded[28]。
Orthonormal mode
Another general application of matrices in physics is to describe linear coupled harmonic systems.Of such systemsEquation of motionIt can be expressed in the form of a matrix, that is, a mass matrix is multiplied by a generalized velocity to give the motion term, and the force matrix is multiplied by the displacement vector to describe the interaction.The best way to find the solution of the system is to find the eigenvector of the matrix (byDiagonalization, called systematicOrthonormal mode。This solution is very important when studying the internal dynamics mode of molecules: the vibration of atoms bound by chemical bonds in the system can be expressed as the superposition of normal vibration modes[29]。When describing mechanical vibration or circuit oscillation, it is also necessary to use normal mode to solve[30]。
geometrical optics
staygeometrical optics You can find many places where you need to use matrices.Geometric optics is a kind ofWave motion of light waveThe approximation theory of the theory, whose model treats light as geometryradial。useParaxial approximation(English:paraxial approximation）, if the angle between the light and the optical axis is very small, thenlensOr the effect of reflection elements on light can be expressed as the product of 2 × 2 matrix and vector.The two components of this vector are the geometric properties (Slope, between light and optical axisMain plane(English:principal plane）Vertical distance).This matrix is calledLight transmission matrix(English:ray transfer matrix）The inner element encodes the properties of the optical element.For refraction, this matrix is subdivided into two types: "refraction matrix" and "translation matrix".The refraction matrix describes the refraction behavior of the ray when it encounters the lens.The translation matrix describes the translation behavior of rays propagating from one main plane to another.
By a series oflensOr the optical system composed of reflective elements can simply describe its ray propagation path with the corresponding matrix combination[31]。
